Monoesters of phosphonic acids



bromo-phenyl or nitro-phenyl, e.g.,

United States Patent Ofitice 3,350,472 Patented Get. 31, 196i 3,350,478 MQNOESTERS F PHOSPHONIC ACIDS Emile Cherbuliez and Joseph Rabinowitz, Geneva, Switzerland, assignors to Hooker Chemical Corporation, Niagara Falls, N.Y., a corporation of New York No Drawing. Filed Nov. 24, 1965, Ser. No. 509,642 Claims. (Cl. 260-953) This is a continuation-impart of United States patent application Ser. No. 230,315, filed Oct. 11, 1962, now U.S. Patent No. 3,268,629.

The present invention relates to new phosphonic monoesters and to an improved process for making phosphonic monoesters. According to the invention, an alcohol is reacted with an anhydride of a phosphonic acid to produce phosphonic monoesters of high purity in very good yields.

As anhydrides of phosphonic acids there are preferably used those commonly known as phosphonic oxides. These example, by the action of the dichloride of a phosphonic acid according to the equation:

0 01 HO 0 0 0 o R l \LR=R l R 21101 in which R represents an aliphatic, aromatic or heterocyclic radical, usually of 1 to 18 carbon atoms, most often of 2 to carbon atoms; these radicals may be substituted, as for example, with a nitro group or a halogen atom, such as fluorine. The radical R may more especially be represented by a monocyclic aryl radical, such as the phenyl, halogeno-phenyl, e.g., p-fluoro-phenyl, p-chloro-phenyl, pp-nitro-phenyl radical.

As a phosphonating agent, one may also use polyphosphonic acids, that is, the partial anhydrides of phosphonic acids, which give a lesser yield of ester, but which permit the recovery, in the form of alkaline earth metal phosphonate, of the phosphonic acid which did not react and which may thus be used for another charge. The polyphosphonic acids are considered to be phosphonic anhydrides.

The polyphosphonic acids are obtained by heating corresponding phosphonic acids at a temperature between about 120 to 200 degrees centigrade and under vacuum (about 3 to 30 millimeters of mercury) for a period of about 48 to 96 hours. The pyrophosphonic acids may also be prepared by the action of a given quantity of water on corresponding dichlorides.

It is sufficient to mix an equivalent of phosphonic oxide (RPO with 0.5 to 3 moles, especially with 1.2 to 3 moles of anhydrous R OH alcohol (excess alcohol) and to heat the mixture for a period of 3 to 48 hours at temperatures of between 20 and 200 degrees centigrade, preferably between 50 and 150 degrees centigrade, and the reaction may also be carried out under vacuum. The mixture becomes homogeneous and the phosphonic mono ester is obtained, as exemplified by the following equation:

Generally, the phosphonic monoester is separated from the mixture in the form of its salt which may be characterized by the formula In the above formulas, R has the significance indicated previously, R represents aliphatic radicals such as alkyl,

and alakaline earth metals, and n is greater than one These radicals, including those which contain double or triple bonds, remain intact during the reaction. The straight chain saturated aliphatic radicals usually are 01 1 to 18 carbon atoms, preferably from 1 to 8 carbon atoms, the straight chain unsaturated aliphatic radicals usually are of 2 to 18 carbon atoms, preferably from 2 to 8 carbon atoms, while the cycloaliphatic radicals are from 3 to 18, preferably from 3 to 12. The radicals R may also carry one or more compatible substituents such as halogen atoms, particularly chlorine or fluorine, free or esterified hydroxyl or carboxyl groups, carboxamide, nitrile or amino groups, preferably tertiary-substituted amino groups of nitrogen atoms, for example, aliphatic radicals such as methyl or ethyl groups or aromatic or such as a phenyl or benzyl radical.

tion of the indicated radicals with respect be phosphonated.

Thus, this reaction is a general one and applies to various alcohol types. If a polyol is treated by a phosphonic oxide, only one functional OH is phosphonated, whatever the quantity of oxide is used. Among the polyols are glycerol, glycol, pentaeryth-ritol, neopentyl glycol, 1,4 (hydroxymethyl)benzene, sorbitol and trimethylol propane. With the halogenated alcohols and particularly with the chlorinated and fluorinated alcohols, the reaction is carried out quite normally and the halogen is retained. With the amino alcohols this reaction proceeds quite well when the amino function is tertiary. Despite the fact that the phenyl-phosphonic oxide has a tendency to react with amino alcohols with unreacted primary or secondary amino groups and only a minimal percentage of phosphonic amide. This amidized group may, if desired, be readily hydrolyzed in an acid medium when the ester portion is relatively stable. The phosphonation reaction is also readily accomplished and in good yield with unsaturated alcohols having double or triple bonds, singly or in multiple. The reaction is more rapid with primary than with secondary alcohols, thus, heating for a longer time will be desirable during phosphonation of a secondary alcohol.

A preferred embodiment of this invention is the reaction of an alcohol with an anhydride of a phosphonic acid in the presence of a tertiary base, such as tri-lower alkylamine, e.g., triethylamine or trimethylamine or a cyclic tertiary base, such as pyridine. The addition of an equivalent of the said tertiary base is especially advantageous if the radical R is the radical of a tertiary aliphatic alcohol or the radical of an alcohol sensitive to acid, such as a terpene alcohol.

If the radical R is the radical of an a-nitrilo alcohol, the nitrile group may be hydrolyzed to form an ester of an a-carboxamido alcohol. If the reaction mixture does not contain any water, there may be intermediate formation of a cyclic derivative of the ester, which is easily hydrolyzed with water to form the monoester of the a-carboxamido alcohol. With [3-, and o-nitrilo alcohols the monoesters of the nitrilo alcohols are preferably obtained; the hydrolysis to the carboxamido derivative does not take place.

The monoesters obtained can be isolated as their metal salts. The alkaline or alkaline earth metal salts of phosphonic monoesters are practically neutral in aqueous solution (pH about 6) and can be preserved indefinitely in aqueous solution (at this pH the time of half-hydrolysis Q n a 0.1 molar aqueous solution of ester at 100 degrees :entigrade is greater than 500 hours); the salts of phosphonic monoesters of amino alcohols are alkaline in aqueous solution (pH 105-11) and also are stable at this pH. 7

In order to isolate the alkaline earth metal or alkali metal salts of the phosphonic monoester, including the calcium, barium, strontium, sodium, potassium, lithium, aluminum and magnesium salts, one proceeds as follows:

After cooling, the excess alcohol is distilled (when a large excess is used) under vacuum. The residue is taken up by water and neutralized by a hydroxide of an alkaline earth metal to a pH of 8.2 (turning of phenolphthalein). (In the case of amino alcohols, it is desirable to add hydroxide sufficient to obtain a pH of about 11.) The phosphonic acid which may still be present (formed by hydrolysis of the oxide which may not have reacted, or formed in the course of the reaction when polyphosphonic acid is used) precipitates partially as an alkaline earth it is completely precipitated by the addition of one volume of alcohol. This is filtered and the filtrate which contains the alkaline earth metal salt of the phosphonic monoester is evaporated and dried under vacuum. Generally, the product obtained is pure. If not, it is purified by extraction with boiling acetone (which dissolves the impurities); after cooling, the alkaline earth metal salt of the phosphonic monoester is filtered.

If it is desired to obtain alkali metal salts of these monoesters, the alkaline earth metal salt is dissolved in water and there is added the required quantity of carbonate of an alkali metal. The precipitate of the alkaline earth metal carbonate formed is filtered off and the filtrate is evaporated to dryness under vacuum. The amorphous residue treated by acetone is transformed into a crystalline precipitate which is the alkali metal salt of the monoester. One may adapt the above methods to make other salts.

The alkaline earth metal salts of the phosphonic monoesters of superior aliphatic alcohols or of terpene alcohols are generally insoluble in water or in 50 percent ethyl alcohol. In order to isolate these monoesters, one proceeds preferably as follows:

(a) After cooling, the reaction mass is taken up by ether and neutralized by triethylamine. The triethylammonium salt of the phosphonic acid is filtered ofr", whereas the triethylammoniurn salt of the phosphonic monoester is soluble. The ether and the excess of the tertiary base are distilled, the residue is taken up by ether and the alkali earth metal salt of the phosphonic monoester is precipitated by adding an aqueous solution of the corresponding alkali earth metal halogenide. The precipitate is washed with water and is purified as previously described.

(b) If the reaction is carried out in the presence of a tertiary base such as trimethylamine, triethylamine or pyridine, the reaction mixture is taken up by ether and the phosphonate of the tertiary amine is filtered off. The ether and the excess of the tertiary base are distilled and the residue is taken up by water. The alkaline earth metal salt of the phosphonic monoester is precipitated by adding an aqueous solution of an alkali earth metal halogenide.

If the free phosphonic monoester crystallizes easily, the reaction mixture is taken up by water and the free phosphonic monoester is precipitated by adding a strong acid, such as hydrochloric acid (this is the case, for instance, of menthyl phenylphosphonic acid).

This new process of phosphonation of alcohols by phosphonic oxides and polyphosphonic acids has a number of advantages: it leads straight to monoesters only, it is simple and directly gives pure products in very high yield, calculated as a percentage of the theoretical value. Moreover, the process is quite general and many of the prepared phosphonic monoesters are new.

The monoesters thus obtained and their salts may be used as additives to liquid fuels, such as gasoline, or as products for impregnating fibers so as to make them fireproof, pest-resistant or vermin-proof.

4 The invention is illustrated but not limited by the following examples. All parts are by weight and all temperatures are in degrees centigrade, unless otherwise indicated.

Example 1 140 parts (1 equivalent) of phenylphosphonic oxide (CGH5PO2)D and 48 to 64 parts (1.5 to 2 moles) of anhydrous methanol are heated to 100 deg-recs centigrade (bath temperature). After cooling, the excess alcohol is evaporated under vacuum and the residue is taken up with 500 to 800 parts of H 0. This solution is neutralized by Ca(OH) (or Ba(OH) if one desires to obtain a barium salt of the monoester) up to a pH at which phenolphthalein turns to colored form (about 8.2). Then an equal volume of alcohol is added and the calcium (or barium) phenylphosphonate thus precipitated is filtered. The filtrate, evaporated to dryness, yields the monomethylphenylphosphonate of calcium (or barium) which is generally pure initially. If it is not, the dry residue is dissolved in boiling acetone and, after cooling, the calcium or barium rnonomethylphenylphosphonate is filtered. The yield in pure product,

is 77 percent of theoretical.

Instead of phenylphosphonic oxide, other aryl phosphonic oxides may be employed to obtain corresponding salts of aryl phosphonic monoesters.

Example 2 The phenylphosphonic monoesters (in the form of their calcium or barium salts) of the alcohols mentioned below have been prepared in the same manner as described in Example 1.

The monoester of ethanol of the formula:

0 OM cam-1 0 C211 Yield=84 percent.

The monoester of propanol of the formula:

0 OM com-i 0 CHzCHzCHa Yield=90 percent.

The monoester of isopropanol of the formula:

0 OM 0,H.- I -o OH(CH3)1 Yield=5 6 percent.

The monoester of butanol of the formula:

0 OM CH -i O CHzCHzCHzCHs Yield= percent.

The monoester of isobutanol of the formula:

0 OM com-I O CH(CH3)CH-.1CH3

Yield=60 percent.

The monoester of amyl alcohol of the formula:

0 OM C H -l O CHzCHzCHzCHaCH;

Yield-=71 percent.

In all the above formulas M is Ca or Ba Example 3 The esters described in Examples 1 and 2 were also obtained with the aid of polyphenylphosphonic acid (rt-2) as the phosphonating agent, in the following manner:

amyl ester. These yields are clearly below those obtained ples 1 and 2) but as in the Examples 1 and 2, one may recover all the henylphosphonic acid which is not transformed into an ester, in the form of an alkaline earth metal salt.

Example 4 140 parts (1 equivalent) of phenylphosphonic oxide and 84 to 112 parts (1.5 to 2.0 moles) of propargyl alcohol are heated to 90 degrees centigrade. One then isolates the monopropargylphenylphosphonate of barium or calcium in the manner already described in Example 1. It is illustrated by the structural formula:

and the yield is 70 percent.

By infrared spectroscopicexamination, the presence of the -CEC- group is verified.

In the same manner, there is prepared in 90 percent yield, phenylphosphonic monoester of allyl alcohol /OM CoHs-P=O CH2 CECHz All of these are new products. In the above formulas M=Ca or Ba Example 5 140 parts (1 equivalent) of henylphosphonic oxide and 120 to 161 parts (1.5 to 2.0 moles) of 2-chloroethanol are heated for 16 hours at 120 degrees centigrade (bath temperature). After cooling, the excess'of chlorohydrin is distilled off under vacuum and one isolates the barium salt of the mono(Z-chloroethyl)phenylphosphonic acid.

OH CgH5-P O OOMaOHzCl in a yield of 85 percent.

In a similar manner, except for the temperature and duration of heating (which are shown in parenthesis), the phosphonic monoesters of halogenated alcohols mentioned below are obtained, With yields indicated.

Phenylphosphonic monoester of 3 chloropropanol 1 (100 degrees centigrade, 16 hours) of the formula:

V 8 /OH CflH5-P=O 0 OHzCHzOHzCl Yield=84 percent.

with the oxides (Exam Phenylphosphonic monoester of 4-chlorobutanol-1 degrees centigrade, 16 hours) of the formula:

OM C H P O OCHzCHzCHzCHzCl Yield=65 percent.

Phenylphosphonic monoester of 1,3-difiuoropropanoldegrees centigrade, 48 hours) of the formula:

OOM

0 OH(CH F)z Yield=56 percent.

Phenylphosphonic monoester of 3 -fluoropropanol l (100 degrees centigrade, 48 hours) of the formula:

OOM

OCHZCHBCHZF Yield=40 percent.

Phenylphosphonic monoester of 2,2,3,3-tetrafluoropropanol-l (100 degrees centigrade, 48 hours) of the formula:

o CHz-CFz-CHF; Yield=60 percent.

Example 6 parts (1 equivalent) of henylphosphonic oxide (C H PO and 91 to 124 parts (1.5 to 2.0 moles) of glycol are heated 48 hours at 100 degrees centigrade (bath temperature). There is isolated the calcium or barium salt of the (2-hydroxyethyl-l)phenylphosphonic acid,

0 OH H/ CGHE P O CHZCHZOH in the manner described in Example 1, with a yield of 60 percent.

In a similar manner, except as it relates to the temperature for the tetramethyleneglycol, there are prepared the alkaline earth metal salts of phenylphosphonic monoesters of 'diols mentioned below with the yields indicated:

Phenylphosphonic monoester of propanediol-1,3 of the formula:

ii/ aHs-P OCHQCHrCHaCHaOH Yield=72 percent.

Phenylphosphonic monoester of pentanediol-1,5 of the formula:

OOM

CaH5

OCHzCHzCHnCHzCHzOH Yield=77 percent.

In the case of butanediol-l,4 (tetramethyleneglycol), the temperature should not exceed 60 degrees centigrade, in order to avoid the formation oftetrahydr'ofurane in the phosphonation process.

Also, even if there is used a large excess of a phosthonating agent, only one group OH is phosphonated.

All of these esters are new. In the formulas given M=Ca or Ba Example 7 140 parts (1 equivalent) of phenylphosphonic oxide and 267 to 336 parts (3 to 4 moles) of dirnethylcolamine (dimethylaminoethanol) are heated for 3 hours at 120 degrees centigrade. The excess of dimethylcolamine is distilled off under vacuum and the residue is dissolved in 800 to 1000 parts of water. There is then added an excess of calcium hydroxide in suspension in water with the mixture being agitated for several minutes. After filtering, one volume of alcohol is added to the filtrate to completely precipitate the calcium phenylphosphonate still prescut. This is followed by further filtration and evaporation under vacuum. An amorphous residue is obtained, which, after being treated with boiling acetone, cooled and filtered, yields a precipitate of calcium dirnethylaminoethylphenylphosphonate. The formula of this compound O CHaCHzN (CH3):

and it was obtained in 50 percent yield.

In treating 1-diethylaminopropano1-2 in the above described manner, there is obtained by evaporation under vacuum an aqueous alcoholic solution of calcium (l-diethylaminopr0pyl-2) phenylpho sphon ate,

0 80.5 o ie-l 0 CH-CH:

H2N C2Hr i which is pure. The yield is 30 percent.

Example 8 140 parts (1 equivalent) of phenylphosphonic oxide and 152.5 (2.5 moles) of colamine are heated at 180 degrees centigrade (bath temperature) up to the point Where the mass becomes homogenous. Then a vacuum is created in the flask, which is kept at this temperature overnight. After cooling, the reactive mass is dissolved in 800 to 1000 parts of water with the addition of an excess of calcium hydroxide. This mixture is agitated and an equal volume of ethanol is added. The liquid is filtered and evaporated dry under vacuum. The residue is treated with boiling alcohol and yields, after cooling, a precipitate of calcium (Z-aminoethyl)phenylphosphonate,

CeHr-P OCHflCHzNHa in 40 percent yield.

Example 9 140 parts (1 equivalent) of phenylphosphonic oxide, 79 parts (1 mole) of pyridine and 111 to 148 parts (1.5 to 2.0 moles) of tertiary butanol are heated for one night at 75 degrees centigrade (bath temperature). A calcium or barium salt of t-butylphenylphosphonic acid,

ll/ CeHs-P OC(CHa)a is obtained in a yield of 60 percent by isolation manner described under Example 1.

in the Example 10 One mole of cetylic alcohol and one equivalent of phenylphosphonic oxide are heated for 48 hours at 110 degrees centrigrade. After cooling, the reaction mixture is taken up by ether and four equivalents of triethylamine phosphonate:

cunt-P OCH2(CH2)MCH3 When cetyl alcohol, used as starting material in the above example, is replaced by one of the alcohols of the following formulas: CH (CH CH OH,

CH (CH CH OH or CHF (CF CH OH, and when using the described procedure, the following barium phenylphosphonic monoesters are obtained:

0 OBS/0.5 C H -i O CH2(CH2)0CH3 Yield=55 percent.

0 OBEWJ C.H. i

O CHKCHmCHi Yield=23 percent.

0 OBBM oHa l O CH2(CF2)9CHF$ Yield:55 percent.

Example 11 One mole of nerol, two moles of triethylamine and one equivalent of phenylphosphonic oxide are mixed carefully (the reaction is exothermic and it is necessary to cool). After a few minutes the mixture becomes homogenous. It is then heated at degrees centigrade for 15 hours (with good agitation). After cooling, the reaction mass is taken up by anhydrous ether. The triethylammonium phenyl hosphonate is filtered off and the ether and the excess of triethylamine are evaporated under reduced pressure. The oily residue is dissolved in water and the pH brought to about 5, by adding dilute hydrochloric acid. When an excess of a concentrated aqueous solution of calcium chloride is added to the preceding solution, voluminous precipitate of the calcium salt of the phenylphosphonic monoester appears. This precipitate, which retains most of the unreacted terpene alcohol, is washed with water and then with acetone which dissolves most of the retained alcohol. To achieve the elimination of the retained alcohol, the precipitate is finally washed with ether and then dried in vacuo over P 0 The yield is 30 percent of pure calcium neryl phenylphosphonate C H O PCa In a similar manner, the calcium salts of the phenylphosphonic monoesters of geraniol and farnesol are obtained in 3 1 percent yield each.

Example 12 Example 13 One mole of lactonitrile and one equivalent of phenylphosphonic oxide are heated at 90 degrees centigrade for 46 hours. After cooling, the reaction cold water and is quickly neutralized by barium hydroxide to a pH of 8.2 (turning of phenolphthalein). One

old alcohol is then added to complete the prebarium phenylphosphonate which is filtered off and discarded. The filtrate is evaporated and the residue is treated with acetone, filtered and dried. The dry residue is treated again several times with-acetone in order to eliminate completely any trace of lactonitrile present, and dried in vacuo. The yield of pure barium Z-carbamido-Z-ethyl phenylphosphonate is 89 percent.

The a-nitrilo alcohols are transformed, under the described conditions, into the corresponding carboxamidoalkyl phenylphosphonic monoesters, 6-, 'yand fi-nitrilo alcohols yield the corresponding nitriloalkyl phenylphosphonic monoesters.

In a similar manner are phosphonated the nitrilo alcohols of the following formulas (yield of the barium salt of the monoester in parenthesis): NCCH2OH (35 percent); NCC(CH OH (64 percent); NCCH CH OH (70 percent); NC-CH(CH H OH (34 percent); NCCH CH CH OH (58 percent);

(59 percent). The first two nitrilo alcohols yield the corresponding carboxamidoalkyl phenylphosphonic monoesters and the other nitrilo alcohols (18-, 'yand 6-nitrilo alcohols) yield the corresponding nitriloalkyl phenylphosphonic monoesters.

Example 14 One mole of anhydrous glycerol and one mole of phenylphosphonic oxide are heated at 165 degrees centigrade until the mass liquefies, and then for 20 hours at The crude salt is reduced to a- 10 taken up by cold alcohol or acetone and poured immedi ately into two liters of water. This mixture is neutralizet with barium hydroxide to a pH of 8.2 and diluted witl water to five liters. One volume of alcohol is added anc CBr CH OP(O) (csHs) (O M) CCI CH2 s- 5) 0.5) Yield: 88%

s s s) o.5) 3

Yield: 78%

3 2 s 5) 0.5) Yield: 81%

Example 16 In a similar manner as described in Examples 1 to 15, p-fluorophenylphosphonic monoesters are obtained in good yields when p-fluorophenylphosphonic oxide (a novel oxide) is reacted with the corresponding alcohol.

Using, for example, 'as starting alcohols: methanol, tbutanol, t-amyl alcohol, propane-1,3-diol, 2-chloro-1- ethanol, n-decanol, 2-amino-1-ethanol, Z-dimethylaminol-ethanol, lactonitrile and 2-nitrilo-1-ethanol, the following p-fluorophenylphosphonic monoesters are obtained: RP(O) (OM) (OCH (70%) s)3 (53%) 2 5)( s) M) HOCH CH CH OP(O) (R) (OM) (70% CICH CH O'P(O) (R) (OM) (62%) CH (CH CH OP(O)(R) (OM) (62%) H NCH CH OP(O) (R) (OM) (53 (CH NCH CH OP(O) (R) (OM) (30%) H NCOCH(CH )OP(O) (R) (OM) (38%) NCCH CH OP(O) (R) (OM) (64%) R represents the p-fluorophenyl radical (FC H M represent an equivalent of an alkali earth metal and the yield is indicated in parenthesis.

p-fluorophenylphosphonic oxide reacts with all the other alcohols mentioned in Examples 1 to 15 to yield the corresponding p-fluorophenylphosphonic monoesters.

The p-fluorophenylphosphonic oxide is prepared as follows according to the equation:

/Cl HO (I) O 0 Cl 110 O barium a-glyceryl phenylphosphonate HOCH -CH[OP(O) (C H Ba ]CH OH barium ,B-glyceryl phenylphosphonate.

The determination of the neighboring OH groups by the method of Malaprade indicates that the product is composed of 51.5 percent of barium a-glyceryl phenylphosphonate and of 48.5 percent of the ,H-derivative.

Example 15 Six moles of tribromoethanol and 4 equivalents of phenylphosphonic oxide are heated at 90 degrees centigrade for 44 hours. After cooling,

the reaction mass'is 7 phenylphosphonic oxide and the precipitate (p-fluorophenylphosphonic oxide) filtered oil. The filtrate is left again for tWo days at 5 degrees centigrade and a second amount of p-fiuorois filtered off. The operation is 1 1 repeated until no further precipitation occurs (after con- :entration of the solution to 200 milliliters). The precipitates are washed with benzene and dried in vacuo at 60 degrees centigrade, In this manner, 60 grams of pure pr fluorophenylphosphonic oxide, melting point 109-111 degrees centigrade, are obtained (yield 88 percent).

(C H O FP) -Calculated: F, 12.0 percent; P, 19.6 percent. Equivalent weight, 158.1. Found: F, 12.4 percent; P, 19.3 percent. Equivalent Weight, 160.

What is claimed is: 1. A compound represented by the formula OM RP 0 wherein R is selected from the group consisting of phenyl, halogeno-phenyl and nitro-phenyl, R is alkyl of 1 to 8 carbon atoms carrying one hydroxyl substituent, and M is selected from the group consisting of alkali metals and alkaline earth metals.

2. A compound in accordance with claim 1 wherein R is selected from the group consisting of phenyl and pfluorophenyl and wherein R is alkyl of 1 to 8 carbon atoms.

3. The compound of the formula:

/0 H C5H5-P=O O CH2-CHzO H 4. The compound of the formula:

0 H. C H P=O 10 ocm-oHT-crnorr 5. The compound of the formula:

OCHz-CHr-CHz-CHzOH CHARLES B. PARKER, Primary Examiner. R. L. RAYMOND, Assistant Examiner. 

1. A COMPOUND REPRESENTED BY THE FORMULA 